Simplify and expand the following expression: $ \dfrac{3}{2n - 2}+ \dfrac{3}{5n + 10}+ \dfrac{4}{n^2 + n - 2} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{3}{2n - 2} = \dfrac{3}{2(n - 1)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{3}{5n + 10} = \dfrac{3}{5(n + 2)}$ We can factor the quadratic in the third term: $ \dfrac{4}{n^2 + n - 2} = \dfrac{4}{(n - 1)(n + 2)}$ Now we have: $ \dfrac{3}{2(n - 1)}+ \dfrac{3}{5(n + 2)}+ \dfrac{4}{(n - 1)(n + 2)} $ The least common multiple of the denominators is: $ 10(n - 1)(n + 2)$ In order to get the first term over $10(n - 1)(n + 2)$ , multiply by $\dfrac{5(n + 2)}{5(n + 2)}$ $ \dfrac{3}{2(n - 1)} \times \dfrac{5(n + 2)}{5(n + 2)} = \dfrac{15(n + 2)}{10(n - 1)(n + 2)} $ In order to get the second term over $10(n - 1)(n + 2)$ , multiply by $\dfrac{2(n - 1)}{2(n - 1)}$ $ \dfrac{3}{5(n + 2)} \times \dfrac{2(n - 1)}{2(n - 1)} = \dfrac{6(n - 1)}{10(n - 1)(n + 2)} $ In order to get the third term over $10(n - 1)(n + 2)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{4}{(n - 1)(n + 2)} \times \dfrac{10}{10} = \dfrac{40}{10(n - 1)(n + 2)} $ Now we have: $ \dfrac{15(n + 2)}{10(n - 1)(n + 2)} + \dfrac{6(n - 1)}{10(n - 1)(n + 2)} + \dfrac{40}{10(n - 1)(n + 2)} $ $ = \dfrac{ 15(n + 2) + 6(n - 1) + 40} {10(n - 1)(n + 2)} $ Expand: $ = \dfrac{15n + 30 + 6n - 6 + 40}{10n^2 + 10n - 20} $ $ = \dfrac{21n + 64}{10n^2 + 10n - 20}$